Posts Tagged ‘subset’
Thursday, December 31st, 2009
Overview of the Universe of Discourse
Description
A detailed tutorial on the universe of discourse. Step by step tutorial including several examples of the universe of discourse for reference.
Overview
The universe of discourse is normally just referred to simply as the universe of a set. The universe of discourse can also be the universe of a certain truth set. Basically, it is all possible considerations for a truth set. It is also a set itself, one where many different subsets are taken from. As you can tell, the universe of discourse has different meanings depending on the exact branch of math you are studying. However, all definitions have one thing in common: the universe is a set where many other sets are taken from. Normally it is easy to figure out what the universe of dicourse is based on the context of the problem you are trying to solve.
Tags: considerations, context, discourse, discrete math, element, problem, set, subset, truth, universe, value
Posted in Discrete Math | No Comments »
Tuesday, December 29th, 2009
Introduction to Ordinary Sets
Description
A detailed tutorial on ordinary sets in set theory. Step by step tutorial including several examples of ordinary sets in set theory for reference.
Overview
You may be reading this and asking yourself, what is an ordinary set? An ordinary set is a set where the complete set is not part of the set. This is not the same as a subset, for as we know all sets are subsets of themselves. An example of an ordinary set is the set of all pencils. The set of pencils is not a pencil, so it is considered an ordinary set. However, the set of all thoughts is a thought. So, that set is not ordinary. In general, all sets are ordinary sets except for certain thoughts and concepts.
Tags: concepts, discrete math, element, example, extraordinary, numbers, objects, ordinary, set, subset, theory, thoughts, unusual
Posted in Discrete Math | No Comments »
Thursday, December 10th, 2009
Inverse Image of Sets
Description
A detailed tutorial on the inverse image of sets. Step by step tutorial on the inverse image of sets for reference. Knowledge of the inverse image of sets is important in advanced discrete mathematics courses.
Overview
Say that you have a function f: A –> B. Then, X is a subset of A and Y is a subset of B. The image of X or the image set of X is f(X) = {y belongs to B: y = f(x) for some x belonging to X}. The inverse image of Y is defined as f^-1(Y) = {x belongs to A: f(x) belongs to Y}. The inverse image is simply a reversed form of the image. Often when asked to find the inverse image, it will help to set up a drawing of the image of the function, connecting everything where it needs to go. Then to find the inverse you simply reverse your work.
Tags: a, b, connect, diagram, discrete math, form, function, image, image set, inverse, mapping, picture, reverse, set, subset, x, y
Posted in Discrete Math | No Comments »
Thursday, November 5th, 2009
Introduction to Nested Intervals
Description
A detailed tutorial on nested intervals and the nested interval theorem. Step by step tutorial including several examples of nested intervals for reference.
Overview
Nested intervals means to have one interval (or multiple intervals) inside of another interval. The intervals will get smaller and smaller the more you add, until they will finally dimish entirely. There is a theorem for nested intervals, called the nested interval theorem. It states that if A_n = [a_n, b_n] is a sequence of closed intervals such that A_n+1 is a subset of A_n for all n belonging to the set of natural numbers, then the union over A_n is not an empty set.
Tags: algebra, closed, empty, interval, natural, nested, number, open, sequence, set, subset, theorem
Posted in Algebra | No Comments »
Thursday, November 5th, 2009
Linear Subspaces Explained
Description
A detailed tutorial on linear subspaces and how to identify linear subspaces. Step by step tutorial including several examples of linear subspaces for reference.
Overview
A linear subspace is usually referred to as simply a subspace, when it needs to be distinguished from other types of subspaces. Linear subspaces are also sometimes referred to as vector subspaces. In mathematical terms, to identify a linear subspace, we say that K is a field (or a set, like of real numbers), and V is a vector space over K. Elements of V are vectors and elements of K are scalars. W is said to be a subset of V. If W is a vector space itself, with the same vector space operations as V, then it has a subspace of V.
Tags: algebra, element, field, k, linear, number, operations, real, scalar, set, space, subset, subspace, v, vector, W
Posted in Algebra | No Comments »
Tuesday, November 3rd, 2009
Well-Ordering Principle Explained
Description
A detailed tutorial on the well-ordering principle. Step by step tutorial including several examples of the well-ordering principle for reference.
Overview
The well-ordering principle states that every nonempty subset of the set of all natural numbers has a smallest element. This is possible because the number zero is not included in the set of natural numbers, and therefore cannot appear in a subset of all natural numbers. The well-ordering principle is equivalant to the Principle of Mathematical Induction, but they are proved in different ways and have different sets. Sometimes it is a better idea to use the Well-Ordering Principle, and other times it is a better idea to use the Principle of Mathematical Induction.
Tags: discrete math, element, induction, mathematical, n!, natural, nonempty, number, ordering, PMI, principle, set, smallest, subset, well, well-ordering, WOP
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Overview of Transitive Relations
Description
A detailed tutorial on the property of transitive relations. Step by step tutorial including several examples of transitive relations for reference.
Overview
A transitive relation can be mathematically defined as for all x and y belonging to A, if x R y, then y R x. In this statement, A is a set, and R is a relation of that set. An empty set is considered to be transitive. Since a transitive relation is defined by a conditional sentence, a proof for the transitive property of relations would be written as a direct proof.
Tags: conditional, direct, discrete math, divides, empty, equal, equivalence, great, greater, implies, proof, property, r, relation, set, subset, transitive, x, y, z
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Overview of Reflexive Relations
Description
A detailed tutorial on the property of reflexive relations. Step by step tutorial including several examples of reflexive relations for reference.
Overview
A reflexive relation can be mathematically defined as for all x belonging to A, x R x. In this statement, A is a set, and R is a relation of that set. If the relation is an empty set, then it is not reflexive, unless the set itself happens to be an empty set. When writing a proof for a reflexive relation, you must attempt to prove that (x, x) does not belong to R. If you cannot prove this, then you know that the relation must be reflexive.
Tags: discrete math, divide, empty, equal, equvalence, greater, less, proof, property, r, reflexive, relation, set, subset, x
Posted in Discrete Math | No Comments »
Thursday, October 29th, 2009
Introduction to Equivalence Relations
Description
A detailed tutorial on equivalence relations and how to find them. Step by step tutorial on finding equivalence relations for reference.
Overview
An equivalence relation is a relation that specifies how a set can be split into subsets. Relations can only be considered equivalence relations if they are reflexive, symmetric, or transitive. It is possible for an equivalence relation to be one of these, two of these, or all three of these, If the relation is none of them, then it is not an equivalence relation. An empty set is considered to be an equivalence relation, because it is both symmetric and transitive.
Tags: discrete math, element, empty, equivalence, reflexive, relation, set, subset, symmetric, transitive
Posted in Discrete Math | No Comments »
Tuesday, October 27th, 2009
The Range of Relations
Description
A detailed tutorial on the range of relations. Step by step tutorial including several examples of the range of relations for reference.
Overview
The range of a relation is denoted as Rng(R) and looks like a normal set. For each ordered pair in a relation, there are two endpoints, x and y. The range is the set of all the y endpoints – that is to say, all the endpoints that come second in the ordered pair. If you are taking the range of the inverse of a relation, then that would be all the x endpoints. When writing the range, the notation used is just the normal notation, not the ordered pair notation.
Tags: cartesian, coordinates, discrete math, element, endpoint, ordered pair, range, relations, second, set, subset
Posted in Discrete Math | No Comments »
